Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems

Abstract

The extraction of the Kolmogorov (metric) entropy from an experimental time signal is discussed. Theoretically we stress the concept of generators and that the existence of an expansive constant guarantees that a finite-time series would be sufficient for the calculation of the metric entropy. On the basis of the theory we attempt to propose optimal algorithms which are tested on a number of examples. The approach is applicable to both dissipative and conservative dynamical systems.

Deterministic Prediction of the Entropy Increase in a Sudden Expansion

“…The methods of Powell and Percival [27], Grassberger and Procaccia [28], and Cohen and Procaccia [29] are then applied to the probability distributions for each segment to develop the spectral entropy for the given segment. The spectral entropy (dimensionless) is defined as:…”

Brownian motion and diffusion: From stochastic processes to chaos and beyond

“…This is a special instance of (7). Then to determine the entropies H m (ǫ), very efficient numerical methods are available 42,43 (the reader may find an exhaustive review in Refs. 8,9).…”

“…The data were then processed by means of standard nonlinear time-series analysis tools, i.e. the Cohen-Procaccia method 42 , to compute the ǫ-entropy. This computation shows a power-law dependence h(ǫ) ∼ ǫ −2 .…”

Dynamical estimates of chaotic systems from Poincaré recurrences

“…9 requires the knowledge of the generating partitions, information that is not trivial to be extracted from complex data. The advantage of the theoretical approach used is its simplicity.…”

“…8 K 2 is calculated from the correlation decay and in Ref. 9 by the determination of a generating partition of phase space that preserves the value of the entropy. But while the method in Ref.…”